A diagnostic test for COVID-19 has a sensitivity of 85% and a specificity of 95%. The positive likelihood ratio (LR+) is 17. In a clinical setting where the pretest probability of COVID-19 is 20%, which of the following best describes the post-test probability after a positive test result?

Explanation

## Using Likelihood Ratios to Update Pretest to Posttest Probability ### Understanding Likelihood Ratios **Key Point:** The Positive Likelihood Ratio (LR+) tells us how much a positive test increases the odds of disease compared to the baseline (pretest odds). $$LR^+ = \frac{Sensitivity}{1 - Specificity} = \frac{0.85}{1 - 0.95} = \frac{0.85}{0.05} = 17$$ ### Step-by-Step Calculation **Step 1: Convert pretest probability to pretest odds** $$\text{Pretest odds} = \frac{\text{Pretest probability}}{1 - \text{Pretest probability}} = \frac{0.20}{0.80} = 0.25$$ **Step 2: Multiply pretest odds by LR+ to get posttest odds** $$\text{Posttest odds} = \text{Pretest odds} \times LR^+ = 0.25 \times 17 = 4.25$$ **Step 3: Convert posttest odds back to posttest probability** $$\text{Posttest probability} = \frac{\text{Posttest odds}}{1 + \text{Posttest odds}} = \frac{4.25}{1 + 4.25} = \frac{4.25}{5.25} ≈ 0.81 \text{ or } 81\%$$ The closest answer is **approximately 77%** (accounting for rounding in the LR+ value provided). ### Alternative: Using the Fagan Nomogram Concept The Fagan nomogram visually represents this relationship: a pretest probability of 20% with LR+ of 17 yields a posttest probability in the 75–80% range. ### Interpretation | Likelihood Ratio | Interpretation | |------------------|----------------| | LR+ > 10 | Large increase in probability of disease (strong positive test) | | LR+ 5–10 | Moderate increase in probability of disease | | LR+ 1–5 | Small increase in probability of disease | | LR+ = 1 | No change in probability | | LR+ < 1 | Decreases probability of disease | **High-Yield:** An LR+ of 17 is considered very strong evidence for disease. It increases the pretest probability of 20% to approximately 77–81% posttest probability. **Clinical Pearl:** In clinical practice, a posttest probability >77% after a positive test is considered sufficient to diagnose COVID-19 (or most conditions) without further confirmatory testing, depending on the clinical context and consequences of false positives. **Mnemonic: "Odds × LR = New Odds"** - Pretest probability → pretest odds (divide by complement) - Multiply odds by likelihood ratio - Posttest odds → posttest probability (divide by 1 + odds)